Dimension Reduction for Polynomials over Gaussian Space and Applications.

We introduce a new technique for reducing the dimension of the ambient space of low-degree polynomials in the Gaussian space while preserving their relative correlation structure. As an application, we obtain an explicit upper bound on the dimension of an "-optimal noise-stable Gaussian partition. In fact, we address the more general problem of upper bounding the number of samples needed to "-approximate any joint distribution that can be non-interactively simulated from a correlated Gaussian source. Our results significantly improve (from Ackermann-like to "merely" exponential) the upper bounds recently proved on the above problems by De, Mossel & Neeman [CCC 2017, SODA 2018 resp.] and imply decidability of the larger alphabet case of the gap non-interactive simulation problem posed by Ghazi, Kamath & Sudan [FOCS 2016]. Our technique of dimension reduction for low-degree polynomials is simple and can be seen as a generalization of the Johnson-Lindenstrauss lemma and could be of independent interest.